The Lorentz transformation of three-acceleration is given by
a)
a
x
′
=
a
x
γ
3
μ
3
,
a
y
′
=
a
y
γ
2
μ
2
+
a
x
u
y
v
c
2
γ
2
μ
3
,
a
z
′
=
a
z
γ
2
μ
2
+
a
x
u
z
v
c
2
γ
2
μ
3
[
γ
=
1
1
−
v
2
c
2
,
μ
=
1
−
u
x
v
c
2
]
{\displaystyle {\begin{matrix}a_{x}^{\prime }={\frac {a_{x}}{\gamma ^{3}\mu ^{3}}},\quad a_{y}^{\prime }={\frac {a_{y}}{\gamma ^{2}\mu ^{2}}}+{\frac {a_{x}u_{y}v}{c^{2}\gamma ^{2}\mu ^{3}}},\quad a_{z}^{\prime }={\frac {a_{z}}{\gamma ^{2}\mu ^{2}}}+{\frac {a_{x}u_{z}v}{c^{2}\gamma ^{2}\mu ^{3}}}\\\left[\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ \mu =1-{\frac {u_{x}v}{c^{2}}}\right]\end{matrix}}}
or in vector notation in arbitrary directions
b)
a
′
=
a
γ
2
μ
2
−
(
a
⋅
v
)
v
(
γ
v
−
1
)
v
2
γ
3
μ
3
+
(
a
⋅
v
)
u
c
2
γ
2
μ
3
[
γ
=
1
1
−
v
2
c
2
,
μ
=
1
−
v
⋅
u
c
2
]
{\displaystyle {\begin{matrix}\mathbf {a} '={\frac {\mathbf {a} }{\gamma ^{2}\mu ^{2}}}-{\frac {\mathbf {(a\cdot v)v} \left(\gamma _{v}-1\right)}{v^{2}\gamma ^{3}\mu ^{3}}}+{\frac {\mathbf {(a\cdot v)u} }{c^{2}\gamma ^{2}\mu ^{3}}}\\\left[\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ \mu =1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right]\end{matrix}}}
Equations a) were given by #Poincaré (1905/06) , #Einstein (1907/08) , #Abraham (1908) , #Laue (1908) , #Brill (1909) , while the vector notation b) was given by #Tamaki (1913) .
w:Henri Poincaré (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:[ R 1]
d
ξ
′
d
t
′
=
d
ξ
d
t
1
k
3
μ
3
,
d
η
′
d
t
′
=
d
η
d
t
1
k
2
μ
2
−
d
ξ
d
t
η
ϵ
k
2
μ
3
,
d
ζ
′
d
t
′
=
d
ζ
d
t
1
k
2
μ
2
−
d
ξ
d
t
ζ
ϵ
k
2
μ
3
{\displaystyle {\frac {d\xi ^{\prime }}{dt^{\prime }}}={\frac {d\xi }{dt}}{\frac {1}{k^{3}\mu ^{3}}},\quad {\frac {d\eta ^{\prime }}{dt^{\prime }}}={\frac {d\eta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\eta \epsilon }{k^{2}\mu ^{3}}},\quad {\frac {d\zeta ^{\prime }}{dt^{\prime }}}={\frac {d\zeta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\zeta \epsilon }{k^{2}\mu ^{3}}}}
where
(
ξ
,
η
,
ζ
)
=
u
{\displaystyle \left(\xi ,\ \eta ,\ \zeta \right)=\mathbf {u} }
k
=
γ
{\displaystyle k=\gamma }
ϵ
=
v
{\displaystyle \epsilon =v}
μ
=
1
+
ξ
ϵ
=
1
+
u
x
v
{\displaystyle \mu =1+\xi \epsilon =1+u_{x}v}
w:Albert Einstein (December 1907, published 1908) defined the transformation by restricting himself to the x-component (apparently due to a printing error, Einstein's expression misses a cube in the denominator on the right hand side):[ R 2]
d
2
x
0
′
d
t
′
2
=
d
d
t
{
d
x
0
′
d
t
′
}
β
(
1
−
v
x
0
′
c
2
)
=
1
β
(
1
−
v
x
˙
0
c
2
)
x
¨
0
+
(
x
˙
0
−
v
)
v
x
¨
0
c
2
(
1
−
v
x
˙
0
c
2
)
etc.
{\displaystyle {\frac {d^{2}x_{0}^{\prime }}{dt^{\prime 2}}}={\frac {{\frac {d}{dt}}\left\{{\frac {dx_{0}^{\prime }}{dt'}}\right\}}{\beta \left(1-{\frac {vx_{0}^{\prime }}{c^{2}}}\right)}}={\frac {1}{\beta }}{\frac {\left(1-{\frac {v{\dot {x}}_{0}}{c^{2}}}\right){\ddot {x}}_{0}+\left({\dot {x}}_{0}-v\right){\frac {v{\ddot {x}}_{0}}{c^{2}}}}{\left(1-{\frac {v{\dot {x}}_{0}}{c^{2}}}\right)}}\ {\text{etc.}}}
w:Max Abraham derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:[ R 3]
q
˙
x
′
=
q
˙
x
ϰ
3
(
1
−
β
q
x
)
3
,
q
˙
y
′
=
q
˙
y
ϰ
2
(
1
−
β
q
x
)
2
+
q
y
β
q
˙
x
ϰ
2
(
1
−
β
q
x
)
3
,
(
ϰ
=
1
−
β
2
)
q
˙
z
′
=
q
˙
z
ϰ
2
(
1
−
β
q
x
)
2
+
q
z
β
q
˙
x
ϰ
2
(
1
−
β
q
x
)
3
,
{\displaystyle {\begin{aligned}{\mathfrak {\dot {q}}}_{x}^{\prime }&={\frac {{\mathfrak {\dot {q}}}_{x}\varkappa ^{3}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{3}}},\\{\mathfrak {\dot {q}}}_{y}^{\prime }&={\frac {{\mathfrak {\dot {q}}}_{y}\varkappa ^{2}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{2}}}+{\frac {{\mathfrak {q}}_{y}\beta {\mathfrak {\dot {q}}}_{x}\varkappa ^{2}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{3}}},\quad \left(\varkappa ={\sqrt {1-\beta ^{2}}}\right)\\{\mathfrak {\dot {q}}}_{z}^{\prime }&={\frac {{\mathfrak {\dot {q}}}_{z}\varkappa ^{2}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{2}}}+{\frac {{\mathfrak {q}}_{z}\beta {\mathfrak {\dot {q}}}_{x}\varkappa ^{2}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{3}}},\end{aligned}}}
or simplified using three-vector
p
˙
{\displaystyle {\mathfrak {\dot {p}}}}
q
˙
x
′
=
p
˙
x
,
ϰ
q
˙
y
′
=
p
˙
y
,
ϰ
q
˙
z
′
=
p
˙
z
(
p
˙
=
q
˙
ϰ
3
(
1
−
β
q
x
)
2
+
q
β
q
˙
x
ϰ
3
(
1
−
β
q
x
)
3
)
{\displaystyle {\begin{matrix}{\mathfrak {\dot {q}}}_{x}^{\prime }={\mathfrak {\dot {p}}}_{x},\qquad \varkappa {\mathfrak {\dot {q}}}_{y}^{\prime }={\mathfrak {\dot {p}}}_{y},\qquad \varkappa {\mathfrak {\dot {q}}}_{z}^{\prime }={\mathfrak {\dot {p}}}_{z}\\\left({\mathfrak {\dot {p}}}={\frac {{\mathfrak {\dot {q}}}\varkappa ^{3}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{2}}}+{\frac {{\mathfrak {q}}\beta {\mathfrak {\dot {q}}}_{x}\varkappa ^{3}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{3}}}\right)\end{matrix}}}
w:Max von Laue wrote the transformation in two dimensions x,y as follows:[ R 4]
q
˙
x
′
=
(
c
c
2
−
v
2
c
2
−
v
q
x
)
3
q
˙
x
,
q
˙
y
′
=
(
c
c
2
−
v
2
c
2
−
v
q
x
)
2
(
q
˙
y
+
v
q
y
q
˙
x
c
2
−
v
q
x
)
,
{\displaystyle {\begin{aligned}{\mathfrak {\dot {q}}}_{x}^{\prime }&=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\mathfrak {q}}_{x}}}\right)^{3}{\mathfrak {\dot {q}}}_{x},&{\mathfrak {\dot {q}}}_{y}^{\prime }&=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\mathfrak {q}}_{x}}}\right)^{2}\left({\mathfrak {\dot {q}}}_{y}+{\frac {v{\mathfrak {q}}_{y}{\mathfrak {\dot {q}}}_{x}}{c^{2}-v{\mathfrak {q}}_{x}}}\right),\end{aligned}}}
w:Alexander von Brill wrote the transformation in which the primed frame moves in z-direction while the x-axis is perpendicular:[ R 5]
d
2
x
′
d
t
′
2
=
d
d
t
x
˙
k
−
k
q
z
˙
⋅
1
d
t
′
d
t
=
1
k
2
x
¨
(
1
−
q
z
˙
)
+
q
x
˙
z
¨
(
1
−
q
z
˙
)
3
d
2
z
′
d
t
′
2
=
d
v
z
′
′
d
t
′
=
z
¨
1
−
q
2
3
(
1
−
q
z
˙
)
3
{\displaystyle {\begin{matrix}{\frac {d^{2}x^{\prime }}{dt^{\prime 2}}}={\frac {d}{dt}}{\frac {\dot {x}}{k-kq{\dot {z}}}}\cdot {\frac {1}{\frac {dt'}{dt}}}={\frac {1}{k^{2}}}{\frac {{\ddot {x}}(1-q{\dot {z}})+q{\dot {x}}{\ddot {z}}}{(1-q{\dot {z}})^{3}}}\\{\frac {d^{2}z^{\prime }}{dt^{\prime 2}}}={\frac {d{\mathfrak {v}}_{z'}^{\prime }}{dt'}}={\frac {{\ddot {z}}{\sqrt {1-q^{2}}}^{3}}{(1-q{\dot {z}})^{3}}}\end{matrix}}}
w:Kajuro Tamaki was the first to formulate the transformation as a single three-vector formula:[ R 6]
a
′
=
a
−
1
c
2
[
v
[
v
q
]
]
+
1
β
(
1
−
β
)
v
1
(
v
1
a
)
β
2
{
1
−
1
c
2
(
v
q
)
}
3
{\displaystyle \mathbf {a} '={\frac {\mathbf {a} -{\frac {1}{c^{2}}}\left[\mathbf {v} [\mathbf {vq} ]\right]+{\frac {1}{\beta }}(1-\beta )\mathbf {v} _{1}\left(\mathbf {v} _{1}\mathbf {a} \right)}{\beta ^{2}\left\{1-{\frac {1}{c^{2}}}(\mathbf {vq} )\right\}^{3}}}}
which he split into two parts: the first in the direction of
v
{\displaystyle \mathbf {v} }
a
v
′
=
a
v
−
1
c
2
β
[
v
[
v
q
]
]
v
β
3
{
1
−
1
c
2
(
v
q
)
}
3
a
v
¯
′
=
a
v
¯
−
1
c
2
[
v
[
v
q
]
]
v
¯
β
2
{
1
−
1
c
2
(
v
q
)
}
3
{\displaystyle {\begin{aligned}\mathbf {a} _{v}^{\prime }&={\frac {\mathbf {a} _{v}-{\frac {1}{c^{2}}}\beta \left[\mathbf {v} [\mathbf {vq} ]\right]_{v}}{\beta ^{3}\left\{1-{\frac {1}{c^{2}}}(\mathbf {vq} )\right\}^{3}}}\\\mathbf {a} _{\bar {v}}^{\prime }&={\frac {\mathbf {a} _{\bar {v}}-{\frac {1}{c^{2}}}\left[\mathbf {v} [\mathbf {vq} ]\right]_{\bar {v}}}{\beta ^{2}\left\{1-{\frac {1}{c^{2}}}(\mathbf {vq} )\right\}^{3}}}\end{aligned}}}
↑ Poincaré (1905/06), p. 160
↑ Einstein (1907/08), p. 432
↑ Abraham (1908), pp. 375-376
↑ Laue (1908), p. 840
↑ Brill (1909), p. 210
↑ Tamaki (1913), p. 242
Abraham, M. (1908), Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung; 2. Auflage Brill, A. (1909), Vorlesungen zur Einführung in die Mechanik raumerfüllender Massen Einstein, A. (1908) [1907], "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen (The Collected Papers of Albert Einstein Vol. 2)" , Jahrbuch der Radioaktivität und Elektronik , 4 : 411–462, Bibcode :1905AnP...322..891E , doi :10.1002/andp.19053221004 English translation in "CPAE Vol. 2" .Laue, M. v. (1908), "Die Wellenstrahlung einer bewegten Punktladung nach dem Relativitätsprinzip" , Berichte der Deutschen Physikalischen Gesellschaft : 838–844 Poincaré, H. (1906) [1905], "Sur la dynamique de l'électron", Rendiconti del Circolo Matematico di Palermo , 21 : 129–176 
![{\displaystyle {\begin{matrix}a_{x}^{\prime }={\frac {a_{x}}{\gamma ^{3}\mu ^{3}}},\quad a_{y}^{\prime }={\frac {a_{y}}{\gamma ^{2}\mu ^{2}}}+{\frac {a_{x}u_{y}v}{c^{2}\gamma ^{2}\mu ^{3}}},\quad a_{z}^{\prime }={\frac {a_{z}}{\gamma ^{2}\mu ^{2}}}+{\frac {a_{x}u_{z}v}{c^{2}\gamma ^{2}\mu ^{3}}}\\\left[\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ \mu =1-{\frac {u_{x}v}{c^{2}}}\right]\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd1bc2da9cc7ffcc67dc0c865fc798d033ebe55)
![{\displaystyle {\begin{matrix}\mathbf {a} '={\frac {\mathbf {a} }{\gamma ^{2}\mu ^{2}}}-{\frac {\mathbf {(a\cdot v)v} \left(\gamma _{v}-1\right)}{v^{2}\gamma ^{3}\mu ^{3}}}+{\frac {\mathbf {(a\cdot v)u} }{c^{2}\gamma ^{2}\mu ^{3}}}\\\left[\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ \mu =1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right]\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad2e6b90a912d2755428bd57a2271e440180537)
![{\displaystyle \mathbf {a} '={\frac {\mathbf {a} -{\frac {1}{c^{2}}}\left[\mathbf {v} [\mathbf {vq} ]\right]+{\frac {1}{\beta }}(1-\beta )\mathbf {v} _{1}\left(\mathbf {v} _{1}\mathbf {a} \right)}{\beta ^{2}\left\{1-{\frac {1}{c^{2}}}(\mathbf {vq} )\right\}^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/512407f66d70c1ebe0df09275793b1f87f5d672e)
![{\displaystyle {\begin{aligned}\mathbf {a} _{v}^{\prime }&={\frac {\mathbf {a} _{v}-{\frac {1}{c^{2}}}\beta \left[\mathbf {v} [\mathbf {vq} ]\right]_{v}}{\beta ^{3}\left\{1-{\frac {1}{c^{2}}}(\mathbf {vq} )\right\}^{3}}}\\\mathbf {a} _{\bar {v}}^{\prime }&={\frac {\mathbf {a} _{\bar {v}}-{\frac {1}{c^{2}}}\left[\mathbf {v} [\mathbf {vq} ]\right]_{\bar {v}}}{\beta ^{2}\left\{1-{\frac {1}{c^{2}}}(\mathbf {vq} )\right\}^{3}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05f24f47a1ae1f065e1c56f3fc673d2ca03a78c0)